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| 2003 | Dr. rer. nat., University of Leipzig | | 2003 - 2004 | Postdoc, Institute for Advanced Study, Princeton, NJ, USA | | 2004 - 2005 | Postdoc, ETH Zürich, Switzerland | | 2005 - 2007 | Heinz Hopf Lecturer, ETH Zürich, Switzerland | | 2007 - 2011 | Professor (W2, Bonn Junior Fellow), University of Bonn | | Since 2011 | Professor (W3), University of Leipzig |
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Research Area B
The main focus of my work in this research area is weak solutions of the Euler equations of ideal fluid flow using geometric methods. In [1], we developed a version of convex integration, originally due to Nash and his work on isometric embeddings, which can be used to construct a large class of weak solutions of the Euler equations exhibiting turbulent behaviour. As neither of us had a strong background in fluid dynamics and turbulence to start with, we profited substantially from the interaction with Ch. Doering (Ann Arbor) and Y. Brenier (Nice), who were in Bonn in 2007 as guests of RA B.
This initial interaction with experts from the fluids community was essential for our research programme, which subsequently resulted in [2] and a joint paper with Brenier.
The connection of turbulence to geometry, more specifically to the theory of bending surfaces, resulted in further collaboration within RA B. Motivated by the famous conjecture of Onsager concerning energy conservation for the Euler equations, we looked at the analogous problem of  embeddings in the Nash-Kuiper setting. There is a well-developed theory of bending surfaces from elasticity theory and the calculus of variations, and thus it was natural to cooperate with Sergio Conti in RA B for this project. Thus, we developed a rather general framework for extending convex integration to Hölder-continuous derivatives. Our hope now is to transfer these ideas back to Euler.
The collaboration with C. De Lellis allowed us to attract him to Bonn for an extended stay in May 2009 for a lecture series. | Research Area A
My previous work on compensated compactness and Tartar's conjecture [3] is related to the celebrated conjecture of Morrey regarding quasiconvexity and rank-one convexity. The essence of the conjecture asks for a characterization of possible microstructures arising in planar deformations.
Although this conjecture is within the setting of the calculus of variations, an interesting connection arises with harmonic analysis and singular integral operators via the Beurling-Ahlfors transform. This connection leads to several exciting new developments in the field, in particular a possibly fruitful way to approach Morrey's conjecture via stochastic optimal control. In this direction, which forms a joint interest with Herbert Koch in RA A, we organized a Winter School in February 2010 with T. Iwaniec (Syracuse), M. Struwe (Zürich) and A. Volberg (Michigan), leading experts in the area, in order to jump-start the activity on this research topic in Bonn. The school was highly successful, with over 30 participants.
A. Volberg, one of the pioneers in the approach via stochastic optimal control, stayed in Bonn for 3 weeks for further discussions. Although we haven't achieved definite results so far, further cooperation with Volberg is planned, with the hope of combining techniques from the calculus of variations (laminates) with techniques from stochastic optimal control (martingale transforms).
Furthermore, we were able to win back M. Struwe, who is well-known for delivering lectures with a very high degree of clarity, for another extended stay in June 2010. He delivered a series of lecture on geometric variational problems and quantization. |
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[ 1] Camillo De Lellis, Jr., László Székelyhidi
The Euler equations as a differential inclusion
Ann. of Math. (2) , 170: (3): 1417--1436
2009
DOI: 10.4007/annals.2009.170.1417[ 2] Camillo De Lellis, Jr., László Székelyhidi
On admissibility criteria for weak solutions of the Euler equations
Arch. Ration. Mech. Anal. , 195: (1): 225--260
2010
DOI: 10.1007/s00205-008-0201-x[ 3] Daniel Faraco, László Székelyhidi
Tartar's conjecture and localization of the quasiconvex hull in \Bbb R2Ã? 2
Acta Math. , 200: (2): 279--305
2008
DOI: 10.1007/s11511-008-0028-1[ 4] Kari Astala, Daniel Faraco, Jr., László Székelyhidi
Convex integration and the Lp theory of elliptic equations
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 7: (1): 1--50
2008[ 5] Bernd Kirchheim, Jr., László Székelyhidi
On the gradient set of Lipschitz maps
J. Reine Angew. Math. , 625: : 215--229
2008
DOI: 10.1515/CRELLE.2008.095[ 6] Jr., László Székelyhidi
Counterexamples to elliptic regularity and convex integration
The interaction of analysis and geometry
of Contemp. Math. : 227--245
Publisher: Amer. Math. Soc., Providence, RI
2007
DOI: 10.1090/conm/424/08104 |
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| 2000 | Gibbs Prize, Oxford University, England, UK | | 2008 | Elected member of the Junge Akademie (BBAW / Leopoldina) | | 2010 | Oberwolfach Prize | | 2011 | ERC Starting Grant |
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| 2004 | Analysis and Geometry (in honour of Yu. Reshetnyak), Novosibirsk, Russia | | 2008 | Geometric Analysis, Elasticity and PDE (in honour of John M. Ball), Edinburgh, Scotland, UK | | 2010 | Andrejewski Day, Bonn | | 2011 | Applied Mathematics from Waves to Fluids (in honour of Claude Bardos), University of Nice, France |
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| 2007 | Lecturer, Oxford, England, UK | | 2009 | Associate Professor, EPF Lausanne, Switzerland | | 2010 | Professor (W2), University of Bonn |
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- PhD theses: 2, currently 2
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